Classical Mechanics - Goldstein

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CLASSICAL MECHANICS
THIRD EDITION
Herbert Goldstein
Columbia University
Charles Poole
University ofSouth Carolina
John Safko
University ofSouth Carolina
...
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Addison
\Vesley
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San FranCISCO Bostor New York
Capetown Hong Kong London Madnd MeXico City
Monlreal MUnich Pam Smgapore Sydney Tokyo Toronto
Contents
1 • Survey of the Elementary Principles 1
1.1 Mechanics of a Particle 1
1.2 Mechanics of a System of Particles 5
I .3 Constraint~ 12
1.4 D'Alembert's Principle and Lagrange's Equations 16
1.5 Velocity-Dependent Potentials and the Dissipation Punction 22
1.6 Simple Applications of the Lagrangian Fonnulation 24
2 • Variational Principles and Lagrange's Equations 34
2.1 Hamilton's Principle 34
2.2 Some Techniques of the Calculus of Variations 36
2.3 Derivation of Lagrange's Equations from Hamilton's Pnnciple 44
2.4 Extension of Hamilton's Principle to ~onholonomic Systems 45
2.5 Advantages of a Variational Principle Fonnulation 51
2.6 Conservation Theorems and Symmetry Properties 54
2.7 Energy Function and the Conservation of Energy 60
3 • The Central Force Problem 70
3.1 Reduction to the Equivalent One-Body Problem 70
3.2 The Equations of Motion and First Integrals 72
3.3 The Equivalent One-Dimensional Problem, and
Classification of Orbits 76
3.4 The Virial Theorem 83
3.5 The Differential Equation for the Orbit, and Integrable
Power-Law Potentials 86
3.6 Conditions for Closed Orbits (Bertrand's Theorem) 89
3.7 The Kepler Problem: Inverse-Square Law of Force 92
1.R The Motion in Time in the Kepler Problem 98
3.9 The Laplace-Runge-Lenz Vector 102
3.10 Scattering in a Central Force Field 106
3.11 Transfonnation of the Scattering Problem to Laboratory
Coordinates 114
3.12 The Three-Body Problem 121
v
vi Conterts
4 • The Kinematics of Rigid Body Motion 134
4.1 The Independent Coordinates of a RIgid Body 134
4.2 Orthogonal TranstonnatlOns 139
4.3 Formal Properties of the Transformation Matrix 144
4.4 The Euler Angles 150
4.5 The Cayley-Klein Parameters and Related Quantities 154
4.6 Euler's Theorem on the Motion of a Rigid Body 155
4.7 Finite Rotations 161
4.8 Infinitesimal Rotations 163
4.9 Rate of Change of a Vector 17]
4.10 The Coriolis Effect 174
5 • The Rigid Body Equations of Motion 184
5.1 Angular Momentum and KInetIc Energy ot MotIon
about a Point 184
5.2 Tensors 188
5.3 The Inertia Tensor and the Moment of Inertia 191
5.4 The Eigenvalues of the Inertia Tensor and the Principal
Axis Transformation 195
5.5 Solving Rigid Body Problems and the Euler Equations of
Motion 198
5.6 Torque-free Motion of a Rigid Body 200
5.7 The Heavy Symmetrical Top with One Point Fixed 208
5.8 Precession of the Equinoxes and of Satellite Orbits 223
5.9 Prccession of Systems of Charges in a Magnetic Field 230
6 • Oscillations 238
6.1 Fonnulation of the Problem 238
6.2 The Eigenvalue Equation and the Principal Axis Transfonnation 241
6.3 Frequencies of Free Vibration, and Nonnal Coordinates 250
6.4 Free Vibrations of a Linear Triatomic Molecule 253
6.5 Forced Vibrations and the Effect of Dissipative Forces 259
6.6 Beyond Small Oscillations: The Damped Driven Pendulum and the
Josephson Junction 265
7 • The Classical Mechanics of the
Special Theory of Relativity
7.1 Basic Postulates of the Special Theory 277
7.2 Lorentz Transformations 280
7.3 Velocity Addition and Thomas Precession 282
7.4 Vectors and the Metric Tensor 286
276
Contents
7.5 I-Fonns and Tensors 289
7.6 Forces in the SpeCIal Theory; Electromagnetism 297
7.7 Relativistic Kinematics of Collisions and Many-Panicle
Systems 300
7.8 Relativistic Angular Momentum 309
7.9 The Lagrangian Formulation of Relativistic Mechanics 312
7.10 Covariant Lagrangian Fonnulations 318
7.11 Introduction to the General Theory of Relativity 324
vii
8 • The Hamilton Equations of Motion 334
8.1 Legendre Transfonnations and the Hamilton Equations
of Motion 334
8.2 Cyclic Coordinates and Conservation Theorems 343
8.3 Routh's Procedure 347
8.4 The Hamiltonian Formulation of Relativistic Mechanics 349
8.5 Derivation of Hamilton's Equations from a
Variational Principle 353
8.6 The Principle of Least Action 356
9 • Canonical Transformations 368
9.1 The Equations of Canonical Transformation 368
9.2 Examples of Canorncal Transformations 375
9.3 The Harmonic Oscillator 377
9.4 The Symplectic Approach to Canonical Transformations 381
9.5 Poisson Brackets rnd Other Canonical Invariants 388
9.6 Equations of Motion, Infinitesimal C:monical Transformations, and
Conscrvation Theorems in the Poisson Bracket Formulation 396
9.7 The Angular Momentum Poisson Bracket Relations 408
9.8 Symmetry Groups of Mechanical Systems 412
9.9 Liouville's Theorem 419
10 • Hamilton-Jacobi Theory and Action-Angle Variables 430
10.1 The Hamilton-Jacobi Equation for Hamilton's Principal
Function 430
10.2 The Hannonic Oscillator Problem as an Example of the
Hamilton-Jacobi Method 434
10.3 The Hamilton-Jacobi Equation for Hamilton's Characteristic
Function 440
10.4 Separation of Variables in the Hamilton-Jacobi Equation 444
10.5 Ignorable Coordinates and the Kepler Problem 445
10.6 Action-angle Variables in Systems of One Degree of Freedom 452
Contents
10.7 Action-Angle Variables for Completely Separable Systems 457
10.8 The Kepler Problem in Action-angle Variables 466
11 • Classical Chaos 483
11.1 Periodic Motion 484
11.2 Perturbations and the Kolmogorov-Amold-Moser Theorem 487
11.3 Attractors 489
11.4 Chaotic Trajectories and Liapunov Exponents 491
11.5 Poincare Maps 494
11.6 Henon-Heiles Hamiltonian 496
11.7 Bifurcations, Driven-damped Hannonic Oscillator, and Parametric
Resonance 505
11.8 The Logistic Equation 509
11.9 Fractals and Dimensionality 516
12 • Canonical Perturbation Theory 526
12.1 Introduction 526
12.2 Time-dependent Perturbation Theory 527
12.3 Illustrations of Time-dependent Perturbation Theory 533
12.4 Time-independent Perturbation Theory 541
12.5 Adiabatic Invariants 549
13 • Introduction to the Lagrangian and Hamiltonian
Formulations for Continuous Systems and Fields 558
13.1 The Transition from a Discrete to a Continuous System 558
13.2 The Lagrangian Formulation for Continuous Systems 561
13.3 The Stress-energy Tensor and Conservation Theorems 566
13.4 Hamiltonian Formulation 572
]3.5 Relativistic Field Theory 577
]3.6 Examples of Relanvistic Field Theorie~ 583
13.7 ~oether"s Theorem 589
Appendix A • Euler Angles in Alternate Conventions
and Cayley-Klein Parameters 601
Appendix B • Groups and Algebras 605
Selected Bibliography 617
Author Index 623
Subject Index 625
Preface to the Third Edition
The first edition of this text appeared in 1950, and it was so well received that
it went through a second printing the very next year. Throughout the next three
decades it maintained its position as the acknowledged standard text for the intro-
ductory Classical Mechanics course in graduate level physics curricula through-
out the United States, and in many other countries around the world. Some major
institutions also used it for senior level un~aduate Mechanics. Thirty years
later, in 1980, a second edition appeared which was "a through-going revision of
the first edition." The preface to the second edition contains the following state-
ment: "1 have tried to retain, as much as possible, the advantages ofthe first edition
while taking into account the developments of the subject itself, its position in the
curriculum, and its applications to other fields:' This is the philosophy which has
guided the preparation of this third edition twenty more years later.
The second edition introduced one additional chapter on Perturbation Theory,
and changed the ordering of the chapter on Small Oscillations. In addition it added
a significant amount of new material which increased the number of pages by
about 68%. Thisthird edition adds still one more new chapter on Nonlinear Dy-
namics or Chaos, but counterbalances this by reducing the amount of material in
several of the other chapters. by shortening the space allocated to appendices, by
considerably reducing the bibliography, and by omitting the long lists of symbols.
Thus the third edition is comparable in size to the second.
In the chapter on relativity we have abandoned the complex Minkowski spare
in favor of the now standard real metric. Two of the authors prefer the complex
metric because of its pedagogical advantages (HG) and because it fits in well with
Clifford Algebra formulations ofPhysics (CPP), but the desire to prepare students
who can easily move forward into other areas of theory such as field theory and
general relativity dominated over personal preferences. Some modem notation
such as I-forms, mapping and the wedge product is introduced in this chapter.
The chapter on Chaos is a necessary addition because of the current interest
in nonlinear dynamics which has begun to play a significant role in applications
of classical dynamics. The majority of classical mechanics problems and apph-
cations in the real world Include nonlineantles, and It IS important for the student
to have a grasp of the complexities involved, and of the new properties that can
emerge. It is also important to realize the role of fractal dimensionality in chaos.
New sections have been added and others combined or eliminated here and
there throughout the book, with the omissions to a great extent motivated by the
desire not to extend the overall length beyond that of the second edition. A section
ix
x Preface to the Third Edition
was added on the Euler and Lagrange exact solutions to the three body problem.
In several places phase space plots and Lissajous figures were appended to illus-
trate solutions. The damped driven pendulum was discussed as an example that
explains the workings of Josephson junctions. The symplectic approach was clar-
ified by writing out some of the matrices. The harmonic oscillator was treated
with anisotropy, and also in polar coordinates. The last chapter on continua and
fields was fonnulated in the modern notation introduced in the relativity chap-
ter. The significances of the special unitary group in two dimensions SU(2) and
the special orthogonal group in three dimensions SO(3) WE're presentf'.rl in more
up-ta-date notation, and an appendix was added on groups and algebras. Special
tables were introduced to clarify properties of ellipses, vectors, vector fields and
I-forms, canonical transformations, and the relationships between the spacetime
and symplectic approaches.
Several of the new features and approaches in this third edition had been men-
tioned as possihilities in the preface to the second edition. such as properties of
group theory, tensors in non-Euclidean spaces, and "new mathematics" of theoret-
ical physics such as manifolds. The reference to "One area omitted that deserves
special attention-nonlinear oscillation and associated stability questions" now
constitutes the subject matter of our new Chapter 11 "Classical Chaos." We de-
bated whether to place this new chapter after Perturbation theory where it fits
more logically. or before Perturbation theory where it is more likely to be covered
in class, and we chose the latter. The referees who reviewed our manuscript were
evenly divided on this question.
The mathernaticallevel of the present edition is about the same as that of the
first two editions. Some of the mathematical physics, such as the discussions
of hermitean and unitary matrices, was omitted because it pertains much more
to quantum mechanics than it does to classical mechanics, and little used nota-
tions like dyadics were curtailed. Space devoted to power law potentials, Cayley-
Klein parameters, Routh's procedure, time independent perturbation theory, and
the stress-energy tensor was reduced.. In some cases reference was made to the
second edition for more details. The problems at the end of the chapters were
divided into "derivations" and "exercises:' and some new ones were added.
The authors are especially indebted to Michael A. Unseren and Forrest M.
Hoffman of the Oak Ridge National laboratory for their 1993 compilation of
errata in the second edition that they made available on the Internet. It is hoped
that not too many new errors have slipped into this present revision. We wish to
thank the students who used this text in courses with us, and made a number of
useful suggestions that were incorporated into the manuscript Professors Thomas
Sayetta and the late Mike Schuette made helpful comments on the Chaos chapter,
and Professors Joseph Johnson and James Knight helped to clarify our ideas
on Lie Algebras. The following profes.~ors reviewed the manuscript and made
many helpful suggestions for improvements: Yoram Alhassid, Yale University;
Dave Ellis, University of Toledo; John Gruber, San Jose State; Thomas Handler,
University of Tennessee; Daniel Hong, Lehigh University; Kara Keeter, Idaho
State University; Carolyn Lee; Yannick Meurice, University of Iowa; Daniel
Preface to the Third Edition xi
Marlow, Princeton University; Julian Noble, University of VIrginia; Muhammad
Numan, Indiana University of Pennsylvania; Steve Ruden, University of Califor-
nia. Irvine; Jack Semura, Portland State Univemty; Tammy Ann Smecker-Hane.
University ofCalifornia, Irvine; Daniel Stump, Michigan State University; Robert
Wald, University of Chicago; Doug Wells, Idaho State University.
It bas indeed been an honor for two of us (CPP and ILS) to collaborate as
co-authors of this third edition of such a classic book fifty years after its first ap-
pearance. We have admired this text since we first studied Classical Mechanics
from the first edition in our graduate student days (CPP in 1953 and JLS in 1960),
and each of us used the first and second editions in our teaching throughout the
years. Professor Goldstein is to be commended for having written and later en-
hanced such an outstanding contribution to the classic Physics literature.
Above all we register our appreciation and aclcnolwedgement in the words of
Psalm 19,1:
Flushing, New York
Columbia, South Carolina
Columbia, South Carolina
July,2ooo
HERBERT GOLDSTEIN
CHARLES P. POOLE, JR.
JOHN L. SAFKO
CHAPTER
1 Survey of theElementary Principles
The motion of material bodies fonned the subject of some of the earliest research
pursued by the pioneers of physics. From their efforts there has evolved a vast
field known as analytical mechanics or dynamics, or simply, mechanics. In the
present century the term "classical mechanics" has come into wide use to denote
this branch of physics in contradistinction to the newer physical theories, espe-
cially quantum mechanics. We shall follow this usagE', interpreting the name to
include the type of mechanics arising out of the special theory of relativity. It is
the purpose of this book to develop the structure of classical mechanics and to
outline some of its apphcations of present-day interest in pure physics. Basic to
any presentation of mechanics are a number of fundamental physical concepts,
such as space, time, simultaneity, mass, and force. For the most part, however,
these concepts will not be :m3JY7ed critically here; rather. they will be assumed as
undefined tenns whose meanings are familiar to the reader.
OJ)
1.1 • MECHANICS OF A PARTICLE
T.et r he the radiu~ vector of a particle from some given ongin and v its vector
velOCity:
dr
v=-, .
at
The linear momentum p of the particle is defined as the product of the particle
mass and its velocity:
p=mv. (1.2)
In consequence of interactions with external objects and fields, the particle may
experience forces of various types, e.g., gravitational or electrodynamic; the vec-
tor sum of these forces exerted on theparticle is the total force F. The mechanics
of the particle is contained in Newton's second law oj motion. whlch states that
there exist frames of reference in which the motion of the particle is described by
the dIfferential equation
F= dp =P.
dt
(1.3)
1
(104)
2 Chapter 1 Survey of the Elementary Principles
or
d
F= -(~v).
dt
In most instances, the mass of the particle IS constant and Eq. (104) reduces to
dv
F=m-=ma,
dt
where a is the vector acceleration of the particle defined by
d2r
a = dt2 '
(I 5)
(1.6)
The equation of motion is thus a differentiJ1 equation of second order, assumiTlg
F does not depend. on higher-order derivatives.
A reference frame in which Eq. (1.3) is valid is called an inertial or Galilean
system. Even within classical mechanics the notion of an inertial system is some-
thing of an idealization. In practice, however, it is usually feasible to set up a co-
ordinate system that comes as close to the desired properties as may be required.
For maTlY purposes. a reference frame fixed in Earth (the "laboratory system") is
a suffiCIent approximation to an inertial sysrem, while for some astronomical pur-
poses it may be necessary to construct an inertial system by reference to distant
galaxies.
Many of the important conclusions of mechanics can be expressed in the form
of conservation theorems: which indicate under what conditions various mechan-
ical quantities are constant in time. Equation (1.3) directly furnishes the first of
these, the
Conservation Theorem/or the Linear Momentum ofa Particle: lfthe totalforce,
F, is lero. then p= 0 and the linear momel'tum, p, is conserved.
The angular momentum of the particle about point O. denoted by L, is defined
as
L = r xp, (1.7)
where r is the radius vector from 0 to the particle. Notice that the order of the
factors is important. We now define the moment offorce or torque about 0 as
N=rxF. (I.R)
The equation analogous to (1.3) for N is obtained by forming the cross product of
r with Eq. (1.4):
d
r x F = N = r x -(mY).
dt
(1.9)
(1.12)
(1.13)
1.1 Mechanics of a Particle
Equation (1.9) can be written in a different form by using the vector identity:
d d
dt(rxmv)=vxmv+rx dt(mv), (1.10)
where the first term on the right obviously vanishes. In consequence of this iden-
tity, Eq. (1.9) takes the fonn
d dL.
N = -(r x my) = - = L. (1.11)
dt dt
Note that both Nand L depend on the point 0, about which the moments are
taken.
As was the case for Eq. (1.3), the torque equation, (1.11), also yields an iJrune-
diate conservation theorem, this time the
Conservation Theorem for the Angular Momentum of a Particle: If the total
torque, N. if zprt) thpn t = O. and the angular momentum L is conserved.
Next consider the work done by the external force F upon the particle in going
from point 1 to point 2. By definition, this work is
Wl2 = (2 F 0 ds.
. I
For constant mass (a~ will be assumed from now on unless otherwise speCIfied),
the integral in Eq. (1.12) reduces to
J
Fods=mJdV ovdt= m f!!..-(V2)dt,
dt 2 dt
and therefore
m 2 2W12 = "2(V2 - Vl)'
The scalar quantity mv2 j2 is called the kinetic energy of the particle and is de-
noted by T, so that the work done is equal to the change m the kinetic energy:
(1.14)
If the force field is such that the work W12 is the same for any physically
possible path between points I and 2, then the force (and the system) is said to be
conservative. An alternative description of a conservative system is obtained by
imagining the particle being taken from point I to point 2 by one possible path
and then being returned to point 1 by another path. The independence of W12 on
the particular path implies that the work done around such a closed circuit is zero,
i.e.
(1.15)
4 Chapter 1 Survey of the Elementary Principles
Physically it is clear that a system cannot be conservative if friction or other dis-
sipation forces are present, because F • ds due to friction is always positive and
the integral cannot vanish.
Ry a Wl:'ll-known theorem of vector analysis, a necessary and sufficient condi-
tion that the work, W12, be independent of the physical path taken by the particle
is that F be the gradient of some scalar function of position:
F = -VV(r), (1.16)
where V is called the potential, or potential energy. The eXistence of V can be
rnferred mtuIDvely by a simple argument. If W12 is independent of the path of
integration between the end points 1 and 2. it should be possible to express WIZ
as the change in a quantity that depends only upon the positions of the end points.
This quantity may be deSignated by - V, so that for a differential path length we
have the relation
F· ds = -dV
or
av
Ff = --~-,
os
which is equivalent to ~. (1.16). Note that in Eq. (1.16) we can add to V any
quantity constant in space, without affecting the results. Hence the zero level ofV
is arbitrary.
For a conservative system, the work done by the forces is
W12 = VI - Vz.
Combining Eq. (1.17) with Eq. (1.14), we have the result
Tl + VI = T2 + V2,
which states in symbols the
(1.17)
(1.18)
En.elgy CUfl.,ervutiun Theorem for a Particle: If Ihe forces acting on a particle
are conservative, then the total energy ofthe particle, T + V, is conserved.
The force applied to a particle may in some circumstances be given by the
gradient of a scalar function that depends explicitly on both the position of the
particle and the time. However, the work done on the particle when it travels a
distance ds,
av
F·ds=--dsas '
is then no longer the total change in - V during the displacement, since V also
changes explicitly with time as the particle moves. Hence. the work done as the
1.2 tv\echanics of a System of Particles 5
particle goes from point 1 to point 2 is no longer the difference in the function V
between those points. While a total energy T + V may still be defined, it is not
conserved during the course of the particle's motion.
1.2 • MECHANICS OF A SYSTEM OF PARTICLES
In generalizing the ideas of the previous section to systems of many particles,
we must distinguish between the external forces acting on the particles due to
sources outside the system. and internal forces on, say, some particle i due to all
other particles in the system. Thus, the equation of motion (Newton's second law)
for the ith particle is written as
"F F(e) .~ JI + , =P,.
J
(119)
where F~e) stands for an external force, and F}I is the internal force on the ith
particle due to the jth particle (F", naturally, is zero). We shall assume that the
F ,} (like the F~e) obey ~ewton's third law of motion in its original form: that the
forces two particles exert on each other are equal and opposite. This assumption
(which does not hold for all types of forces) is sometimes referred to as the weak
law ofaction and reactzon
Summed over all particles, Eq. (1.19) takes the form
d
2
" ~ (e) '"
d 2 ~m,r, = ~F, + L"FJ,.t .
, I ',J
i:l:-j
(1.20)
(1.21)
The first sum on the right is simply the total external force F(e) , while the second
tenn vanishes, since the law of action and reaction states that each pair F,j +F J1
is zero. To reduce the left-hand side, we define a vector R as the average of the
radli vectors of the particles, weighted in proportion to their mass:
R _ Lmjr, = Lm,rj .
Lm,· M
The vector R defines a pomt known as the center ofmaJs, or more loosely as the
center of gravity, of the system (cf. Fig. 1.1). With this definition, (1.20) reduces
to
(1.22)
which states that the center of mass moves as if the total external force were
acting on the entire mass of the system concentrated at the center of mass. Purely
internal forces, if the obey Newton's thIrd law, therefore have no effect on the
Chapter 1 Survey of the Elementary Principles
• •
o•
•
•
•
'-Center of mass
•
FIGURE 1.1 The center of m~~ of a system of particles.
motion of Lhe center of mass. An oft-quoted example IS the motlon ot an exploding
shell-the center of mass of the fragments traveling as if the shell were still in a
single piece (neglecting air resistance). The same principle is involved in jet and
rocket propulsion. In order that the motion of the center of mass be unaffected,
the ejection of the exhaust gases at high velocity must be counterbalanced by the
forward motion of the vehicle at a slower velocity.
By Eq. (1.21) the totallmear momentum ot" the system,
L dr; dRp= m -=M-, dt dt' (1.23)
is the total mass of the system times the velocity of the center of mass. Conse-
quently, tht> t>quation of motion for the center of mass, (1.23), can be restated us
the
ConselVation Theorem/or the Linear Momentum ofa System ofParticles: lfthe
total external force is zero, the total linear momentum is conserved.
We obtain the total angular momentum of the system by forming the cross
product r, X P, and summing over i. If this operation is performed in Eq. (1.191,
there results, with the aid of the identity, Eq. (1.10),
L(r, x P,) = L :/r, x Pr) = t = Err x F~e) + Lr, x FJi. (1.24)
" i.)
i"!'J
The last tenn on the right in (1.24) can be ~onsidered a sum of the pairs of the
fonn
(1.25)
1.2 Mechanics of a System of Particles
• •
FIGURE 1.2 The vector rij between the ith and jth particle..~.
7
(1.26)
using the equality of action and reaction But rj - r) is identical with the vector
r,j from j to i (cf. Fig. 1.2), so that the right-hand side ofEq. (1.25) can be wntten
as
If the internal forces between two particles, in addition to being equal and oppo-
site, also lie alung lhe line joining the palticles-a condition known as thc strong
law ofaction and reaction-then all of these cross products vanish. The sum over
pairs is zero under this assumption and Eq. (1.24) may be written in the fonn
dL = N(e).
dt
The time derivative of the total angular momentum is thus equal to the moment
of the external force about the given point. Corresponding to Eq. (1.26) is the
Conservation Theoremfor Total Angular Momentum: L is constant in time If the
applied (external) torque is zero.
(It is perhaps worthwhile to emphasize that this is a vector theorem; i.e., L z
will be conserved if Nt> is zero, even if N~e) and Ny> are not zero.)
Note that the conservation of Iinear momentum in the absence of applied forces
assumes that the weak law of action and reaction is valid for the internal forces.
The conservation of the total angular momentum of the system in the absence of
applied torques requires the validity of thc strong law of action and reaction-that
the internal forces in addition be central. Many of the familiar physical forces,
such as that of gravity, satisfy the strong form of the law. But it i~ possible to
find forces for which action and reaction are equal even though the forces are not
central (see below). In a system involvmg moving charges, the forces between
the charges predicted by the Biot-Savart law may indeed violate both forms of
8 Chapter 1 Survey of the Elementary PrrnCiples
the action and reaction law.*Equations (1.23) and (1.26), and their corresponding
conservation theorems, are not applicable in such cases, at least in the form given
here. Usually it is then possible to find some generalization of P or L that is
conserved. Thus, in an isolated system of movin~ charges it is the sum of the
mechanical angular momentum and the electromagnetic "angular momentum" of
the field that is conserved.
Equation (1.23) states that the total linear momentum of the sy~1em is the same
as if the entire mass were concentrated at the center of mass and moving with it.
The analogous theorem for angular momentum is more complicated. With the
ongin 0 as reference point, the total angular momentum of the system is
L=Lr,xp,.
I
Let R be the radius vector from 0 to the center of mass, and let r, be the radius
vector from the center of mass to the t th particle. Then we have (cf. Fig. 1.3)
r, = r, +R (1.27)
and
I
V, = v, + V
where
dR
v=-
dt
Center
of mass
FIGURE 1.3 The vector~ involved in the shIft of reference pomt for the angular momen-
tum.
*If two charge~ are movmg umformly with parallel v~loclty vectois that are not perpendicular to the
line Joining the charge~, then the net mutual forces are equal and opposIte but do not he along the
vector between the charges. ConsIder, further, two charges moving (instanuneou!>ly) ~o as to "cross
Lhe T," I.e., one charge moving diroctly at the other, wluch in turn b moving at right angle~ to the filst
Then the second charge ex.ert~ :i nonvani~htngmdgnetic force on the fir~t, without cxpenencmg my
magnetic reaction force at that mstant.
1.2 Mechanics of a System of Particles
is the velocity of the center of mass relative to 0, and
9
~ = dr:
, dt
is the velocity of the ith particle relative to the center of mass of the system. Using
Eq. (1.27), the total angular momentum takes on the fonn
The last two tenns in this expression vanish, for both contain the factor L mzr; ,
which, it will be recognized, defines the radius vector of the center of mass in the
very coordinate system whose origin is the center of mass and is therefore a null
vector. Rewriting the remaining terms, the total angular momentum about 0 is
L=R xMv+ I:r: xP:.
z
(1.28)
In \\oords, Eq. (1.28) says that the total angular momentum about a point 0 is
the angular momentum of motion concentrated at the center of mass, plus the
angular momentum of motion about the center of mass. The fonn of Eq. (1.28)
emphasizes that in general L depends on the origin 0, through the vector R. Only
if lhe centel of mass l~ at rest with respect to 0 will the angular momentum be
independent of the point of reference. In this case, the first tenn in (1.28) vanishes,
and L always reduces to the angular momentum taken about the center of mass.
Finally, let us consider the energy equation. As in the case of a single particle,
we calculate the work done by all forces in moving the system from an initial
configuration 1, to a final configuration 2:
W12 = L (2 F, • ds, = L (2 F~e) • dsz +L /2 F), • ds,.
i 11 i 11 I.} 1
';f-J
Again, the equations of motion can be used to reduce the rntegrals to
L {2 F, • ds = I: {2 mzv,. vzdt = L {2 d (~miv~).
J 11 J 11 , 11
(1.29)
Hence, the work done can still be written as the difference of the final and initial
kinetic energies:
where T, the total kinetic energy of the system, is
T = ~ Lm,v:.
I
(1.30)
10 Chapter I Survey of the I:lementary Principles
Making use of the transfonnations to center-of-mass coordinates, given in Eq.
(1.27), we may also write T as
1
T = 2: Lml(v+~). (v+~)
,
1 "" 2 1 "" r2 d ("" ,)= 2L.Jm,v + 2L.Jm'Vi + v· dt ~mlr, ,
, I I
and by the reasoning already employed in calculating the angular momentum, the
last tenn vanishes, leaving
(1.31)
The kinetic energy, like the angular momentum, thus also consists of two parts:
the lanetlc energy obtained if all the mass v.ere concentrated at the center of mass,
plus the kinetic energy of motion about the center of mass.
Consider now the right-hand side of Eq. (1.29). In the ~pecial case that the
external forces are derivable in terms of the gradient of a potential, the first term
can be written as
12 r2 12L F~e) • ds, = - L J1 VI V; • ds, = - L V, ,I 1 I I , 1
where the subscript i on the del operator mdicates that the derivatives are with
respect to the components of r,. If the internal forces are also conservative, then
the mutual forces between the ith and jth particles, Fil and F 11, can be obtained
from a potentlal function V,}, To satisfy the strong law of action and reaction, l-~J
can be a function only of the distance betweenthe particles:
(1.32)
The two forces are then automatically equal and opposite,
(1.33)
and lie along the line joining the two particles,
(1.34)
where j is some scalar function. If 'Y,J were also a function of the difference of
some other pair of vectors associated with the particles, such as their velocities
or (to step into the domain of modem phYSICS) their intrinsic "spin" angular mo-
menta, then the forces would still be equal and opposite, but would not necessarily
lie along the direction between the particles.
1.2 Mechanics of a System of Particles 11
When the forces are all conservative, the second tenn m Eq. (1.29) can be
rewritten as a sum over pairs of particles, the tenns for each pair being of the
fonn
-[\VI Vij • ds, +Vj V,] • dSi)'
If the difference vector r, - r j is denoted by r
'j , and if 'Vij stands for the gradient
with respect to r ,J , then
and
so that the term for the ~. pair has the fonn
- f V
'j V,} • dr,j •
The total work arising from internal forces then reduces to
(1 35)
The factor! appears in Eq. (1.35) because in summing over both i and j each
member of a given pair;s included twice, first in the i summation and then in the
.1 summation.
From these considerations, it is clear that it the external and mtemal forces are
both derivable from potentials it is possible to define a total potential energy, V,
of the system,
(1.36)
such that the total energy T + V is conserved, the analog of the conservation
theorem (1.18) for a single particle.
The second tenn on the right in Eq. (1.36) will be called the internal potential
energy of the system. In general, it need not be zero and, more important, it may
vary as the system changes with time. Only for the particular class of system,,:
known as rigid bodie:, will the internal potential always be constant. Formally,
a rigid body can be defined as a system of particles in which the distances r
'j
are fixed and cannot vary with time. In such case, the vectors drij can only be
perpendicular to the corresponding r'j' and therefore to the Fij' Therefore, m a
rigid body the internal jorces do no work, and the internal potential must remain
12 Chapter 1 ~urvey of the Elementary Principles
constant. Since the total potential is in any case uncertain to within an additive
constant. an unvarying internal potential can be completely disregarded in dis~
cussing the motion of the system.
1.3 • CONSTRAINTS
From the previous sections one might obtain the impression that all problems in
mechanics have been reduced to solving the set of differential equations (1.19):
•• F(e) '"'Fm,r, = , - L JlO
J
One merely substitutes the various forces acting upon the particles of the system.
tum.~ the mathematical crank, and grinds out the answers! Even from a purely
physical standpoint. however, this view is oversimplified. For example, it may be
necessary to take Into account the constraints that limit the motion of the system.
We have already met one type of system involving constraint!>, namely rigid bod-
ies, where the constraints on the motions of the particles keep the distances r'J
unchanged. Other examples of constrained systems can easily be furnished. The
beads of an abacus are constrained to one-dimensional motion by the supporting
wires. Gas molecules within a container are constrained by the walls of the ves-
sel to move only inside the container. A partIcle placed on the surface of a solid
sphere is subject to the constraint that it can move only on the surface or in the
region exterior to the sphere.
Constraints may be classified in various ways, and we shall use the following
system. If the conditions of constraint can be expressed as equations connecting
the coordinates of the particles (and possibly the time) having the fonn
f(rl. rz. r3,···, t) = 0, (1.37)
then the constraints are said to be holonomic. Perhaps the simplest example of
holonomic constraints is the rigid body, v"here the constraints are expressed by
equations of the form
A partkle constrained to move along any curve or on a given surface is another
obvious example of a holonomic constraint, WIth the equations defining the curve
or surface acting as the equations of a constraint.
Consuaints nul expr~iblein this fashion are called nonholonomic. The walls
of a gas container constitute a nonholonomic constraint. The constraint involved
in the example of a particle placed on the surface of a sphere i~ also nonholo-
nomic, for it can be expressed a~ an inequality
1.3 Constramts 13
(where a is the radius of the sphere), which is not in the form of (1.37). Thus, in
a gravitational field a particle placed on the top of the sphere will slide down the
surface part of the way but will eventually fall off.
Constraints are further classified according to whether the equations of con-
straint contain the time as an explicit variable (rheonomous) or are not expbcltly
dependent on time (scleronomous). A bead sliding on a rigid curved wire fixed
in !o.pace is obviously subject to a scleronomous constraint; if the wire is moving
in SClme prescribed fashion, the constramt is rheonomous. Note that if the wire
moves, say, as a reaction to the bead's motion, then the tune dependence of the
constraint enters in the equation of the constraint only through the coordinates
of the curved wire (which are now part of the system coordinates). The overall
constraint is then 5>cJeronomous.
Constraints introduce two type..; of difficulties in the solution of mechanical
problems. First, the coordinates r, are no longer all independent, since they are
connected by the equations of constraint; hence the equations of motion (1.19)
are not all independent. Second, the forces of constraint, e.g., the force that the
wire exerts on the bead (or the wall on the gas particle), is not furnished a pri-
ori. They are among the unknowns of the problem and must be obtained from the
solution we s.eek. Indeed, imposing constraints on the system IS simply another
method of stating that there are forces present in the problem that cannot be spec-
Ified directly but are known rather in term~ of their effect on the motion of the
system.
In the case of holonomic constraints, the first difficulty is solved by the intro-
duction of generalized coordinates. So far we have been thinking implicitly in
terms of Cartesian coordinates. A system of N particles. free from constraint~,
has 3N independent coordinates or degrees offreedom. If there exist holonomic
constraints, expressed in k equations in the form (1.37), then we may use these
equations to eliminate k of the 3N coordinates, and we are left with 3N - k inde-
pendent coordinates, and the system is said to have 3N - k degrees of freedom.
Thi~ elimination of the dependent coordinates can be expressed in another way,
by the introduction of new, 3N - k, independent variables Ql, q2, ... , q3N-k in
tenns of which the old coordinate~ rl, f2, ... ,rN are expressed by equations of
the Conn
(1.38)
containing the constraints in them implicitly. These are transformation equations
from the set of (rt) variables to the (qL) set, or alternatively Eqs. (1.38) can be con-
sidered as parametric representations of the (rt) variables. It is always assumed
that we can also transfonn back fLOm the (qt) to the (rl) set, Le., that Eqs. (1.38)
combined with the k equations of constraint can be inverted to obtain any ql as a
function of the (rt) variable and time.
14 Chapter 1 ~urvey ot the Elementary Prmciples
Usually the generalized coordinates, qt, unlike the Cartesian coordinates, will
not divide into convenient groups of three that can be associated together to fonn
vectors. Thus, in the case of a particle constrained to move on the surface of a
sphere, the two angles expressing position on the sphere, say latitude and longi-
tude, are obvious possible generalized coordinates.Or, in the example of a double
pendulum moving in a plane (two particles connected by an inextensible I1ght
rod and suspended by a similar rod fastened to one of the particles), satisfactory
generalized coordinates are the two angles ~h, fh. (Cf. Fig. 1.4.) Generalized co-
ordinates, in the sense of coordinates other than Cartesian, are often useful in
systems without constraints. Thus, in the problem of a particle moving in an ex-
ternal central force field (V = V (r)), there is no constraint involved, but It is
clearly more convenient to use spherical polar coordinates than Cartesian coordi-
nates. Do not, however, think of generalized coordinates in terms of conventional
orthogonal position coordinates. All sorts of quantities may be impressed to serve
as generalized coordinates. Thus, the amplitudes in a Fourier expansion of r J may
be used as generalized coordinates, or we may find it convenient to employ quan-
tities with the dimensions of energy or angular momentum.
If the constraint is nonholonomic, the equations expressing the constraint can-
not be used to eliminate the dependent coordinates. An oft-quoted example of
a nonholonomic constraint is that of an object rolling on a rough surface with-
out slipping. The coordinates used to describe the system wiJl generally involve
angular coordinates to specify the orientation of the body. plus a c;;et of coordi-
nates describing the location of the point of contact on the surface. The constraint
of "rolling" connects these two sets of coordinates; they are not independent. A
change in the position of the point of contact inevltably means a change in its
orientation. Yet we cannot reduce the number of coordinates, for the "rolling"
condition is not expressible as a equation between the coordinates, in the manner
of (1.37). Rather. it is a condition on the vplncities (i e • the point of contact is
stationary), a differential condition that can be given in an integrated fonn only
after the problem is soh.ed.
FIGURE 1.4 Double pendulum.
I oJ Constra Ints
z
o x
FIGURE 1.5 Vertical disk rolhng on a horizontal plane.
15
A simple case will illustrate the point. Consider a disk rolling on the horizontal
xy plane constrained to move so that the plane of the disk is always vertical.
The coordinates used to describe the motion might be the x. y coordinates of the
center of the disk, an angle of rotation ¢ about the axis of the disk, and an angle
() between the axis of the disk and say, the x axis (cf. Fig 1.5). As a result of the
co~straint the velocity of the center of the disk, v, has a magnitude proportional
to¢,
v=a¢,
where a is the radius of the disk, and its direction is perpendicular to the axis of
the disk:
i = v sin 8
y = -v cosO.
Combining these conditions, we have two differential equations of constraint:
dx - a sined¢ = 0,
dy +a cos6d¢ = O.
( 1.39)
Neither of Eqs. (1.39) can be integrated without in fact solving the problem; i.e.,
we cannot find an integrating factor f (x, ), 0, ¢) that will turn either of the equa-
tions into perfect differentials (cf. Derivation 4).* Hence, the constraints cannot
be reduced to the fonn of Eq. (1.37) and are therefore nonholonomic. Physically
we can see that there can be no direct functional relation between ¢ and the other
coordinates x, y, and 8 by noting that at any point on its path the disk can be
*In prin::lple, an Integratmg factor can always be found for a first-order di:Terential equatIon of con-
~traint In systems involving only two coordinates and such constraints are therefore holonomic. A
familiar example IS the two-dimensional motion of a CIrCle rolling on an inclmed plane.
16 Chapter 1 ~urvey ot the Elementary PnnClples
made to roll around in a circle tangent to the path and of arbitrary radius. At the
end of the process, x, }, and () have been returned to their original value~, but cP
has changed by an amOJnt depending on the radius of the circle.
Nonintegrable differential constraint,;; of the fonn of Eqs. (1.39) are of course
not the only type of nonholonomic constraints. The constraint conditions may
involve higher-order derivatives, or may appear in the fonn of inequalities, as we
have seen.
Partly because the dependent coordinates can be eliminated, problems involv-
ing holonomic constraints are always amenable to a fonnal solution. But there is
no general way to attack nonholonomic examples. True, if the constraint is nonin-
tegrable, the differential equations of constraint can be introduced into the prob-
lem along with the differential equations of motion, and the dependent equations
eliminated, in effect, by the method of Lagrange multipliers.
We shall return to this method at a later point. However, the more vicious cases
of nonholonomic constraint must be tackJed individually, and consequently in the
development of the more formal aspects ofclassical mechanics, it is almost invari-
ably assumed that any constraint, if present, is holonomic. This restriction does
not greatly limit the applicability of the theory, despite the fact that many of the
constraints encountered in everyday life are nonholonomlc. The reason is that the
entire concept of constraints imposed in the system through the medium of wires
or surfaces or walls is particularly appropriate only in macroscopic or large-scale
problems. But today physicists are more interested III atomic and nuclear prob-
lems. On this scale all object~, both in and out of the system, consist alike of
molecules, atoms, or smaller particles, exerting definite forces, and the notion of
constraint becomes artificial and rarely appears. Constraints are then used only
as mathematical idealizations to the actual physical case or as classical approxi-
mations to a quantum-mechanical property, e.g., rigid body rotations for "spin."
Such constraints are always holonomic and fit smoothly into the framework of the
theory.
To surmount the second difficulty, namely, that the forces of constraint are
unknown a priori, we should like to so formulate the mechanics that the forces
of constraint disappear. We need then deal only with the known applied forces. A
hint as to the procedure to be followed is provided by the fact that in a particular
system with constraints i e a rigid body, the work done by internal forces (which
are here the forces ot constraint) vanishes. We shall follow up this clue in the
ensuing sections and generalize the ideas contained in it.
1.4 • D'ALEMBERT'S PRINCIPLE AND LAGRANGE'S EQUATIONS
A virtual (infinitesimaft displacement of a system refers to a change in the con-
figuration of the system a!. the result of any arbitrary infinitesimal change of the
coordinates 8rr , consistent with the forces and constraints imposed on the sy.~tem
at the given instant t. The displacement is called virtual to distinguish it from an
actual displacement of the system occurring in a time interval dt, during which
1.4 D'AIembert's Principle and Lagrange's Equations 17
the forces and constrdints may be changing. Suppose the system is in equilibrium;
i.e., the total force on each particle vanishes, F, = O. Then clearly the dot product
F, • 8r
"
which is the virtual work of the force F, in the displacement 8rr , also
vanishes. The sum of these vanishing products over all particles must likewise be
zero:
LF, .8r, = O.
i
(lAO)
As yet nothing has been said that has any new physical content. Decompose F~
into the applied force, F~a), and the force of constraint, f r ,
so that Eq. (1.40) becomes
LF~a). ~rr + LF, .~rl = 0
, r
(1.41)
(1 42)
We now restrict ourselves to systems for which the net virtual work of the
forces ofconstraint is zero. We have seen that this condition holds true for rigid
bodies and it is valid for a large number of other constraints. Thus, if a particle is
constrained to move on a surface, the force of constraint is perpendicularto the
surface, while the virtual displacement must be tangent to it, and hence the virtual
work vanishes. This is no longer true if sliding friction forces are present, and
we must exclude such systems from our fonnulation. The restriction is not un-
duly hampering, since the friction is essentially a macroscopic phenomenon. On
the other hand, the forceli of rolling friction do not violate this condition, since the
forces act on a point that is momentarily at rest and can do no work in an infinites-
imal dtsplacement consistent with the rolling constraint. Note that if a particle is
constrained to a surface that is itself moving in time, the force of constraint is
instantaneously perpendlcular to the surface and the work during a virtual dis-
placement is still zero even though the work during an actual displacement in the
time dt does not necessarily vanish.
We therefore have as the condition for equilibrium of a system that the virtual
work of the appliedforees vanishes:
LF~a) ·8rr = O. (1.43)
Equation (1.43) is often called the principle of virtual work. Note that the coef-
ficient~ of Sri can no longer be set equal to zero; Le., in general F~a) I- 0, since
the 8r, are not completely independent but are connected by the constraint~. In
order to equate the coefficients to zero, we must transfonn the principle into a
fonn involving the virtual displacements of the q~, which are independent. Equa-
tion (1.43) satisfies our needs in that it does not contain the f" but it deals only
with statics; we want a condition involving the general motion of the system.
18 Chapter 1 Survey of the Elementary Principles
To obtain such a principle, we use a device first thought of by James Bernoulli
and developed by D'Alembert. The equation of motion,
F, =P"
can be written as
F, -PI = 0,
which states that the particles in the system will be in equilibrium under a force
equal to the actual force plus a "reversed effective force" -Pi. Instead of (l.40),
we can immediately write
L(F; - P,), ~r, = 0, (1.44)
and, making the same resolutIon Into applied torces and forces ofconstraint, there
results
~:).F~a) - Pi) • ~r, +L f, • or; = 0.
i I
We agoin restrict ourselves to systems for which the virtual work of the forces of
constraint vanishes and therefore obtain
L(F~O) - Pi) • or, = 0,
i
(1.45)
which IS often called D'Alembert's principle. We have achieved our aim. in that
the forces of constraint no longer appear. and the superscript (0) can now ~
dropped without ambiguity. It is still not in a useful form to furnish equations
of motion for the system. We must now transform the principle into an expression
involving virtual displacements of the generalized coordinates, which are then in-
dependent of each other (for holonomic constraints), so that the coefficients of the
fJq, c~n be set separately equal to zero.
The translation from r, to qJ language starts from the transformation equations
(1.38),
(1.45')
(assummg n independent coordinates), and is carried out by means of the usual
"chain rules" of the calculus of partial differentiation. Thus, Vi is expressed in
tenns of the qk by the formula
(1.46)
, .4 LJ'Alembert's PrinCiple and Lagrange's I:quatlons 19
Similarly, the arbitrary virtual displacement 8f, can be connected with the virtual
displacements 8q, by
(147)
Note that no variation of time, 8t, is invo~ved here, since a virtual displacement
by definition considers only displacement... of the coordinates. (Only then is the
virtual displacement perpendicular to the force of constraint if the constraint itself
is changing in time.)
In tenns of the generalized coordinates. the virtual work of the F, becomes
'"' '"' Of,L- F; .8r; = L-Fr . -a-8q)
, r>J q)
0.48)
where the Q) are called the components of the generalizedforce, defined as
" or,Q) =L..,Fr .-.
i oqJ
(1.49)
Note that just as the q's need not have the dimensions of length, so the Q's do
not neces!'arily have the dimensions of force, but QJ8qJ mu~t always have the
dimemions of work. Forexample, Q) might be a torque N) and dqJ a differential
angle dB), which makes N) dB) a differential of work.
We tum now to the other other term Involved In Eq. (1.45), which may be
written as
LP, ,8r, = Lm;r; ·8f,.
r
Expressing 8f, by (1.47), this becomes
" .. orr£-m,f, • -8q).
',) aq)
Consider now the relatIon
~ .. or, ~ [d ( . of, ) . d (Ofi)]L-m,f'·-=LJ - mrfr ·- -mrf,·- -- .
, oqJ r dt oq) dt oqJ (1.50)
In the last term of Eq. (1.50) we can interchange the differentiation with respect
to t and q), for, in analogy to (1.46).
20 Chapter 1 Survey of the Elementary Principles
d (a) a· a2 a2- ~ =~=L. ~~ qk+~'dt aQj aq) k aq} aqk oq/)t
OVi
--,
aq)
by Eq. (1.46). Further, we also see from Eq. (1.46) that
aV~ ar~
oq) aqj
Substitution of these changes in (1.50) leads to the result that
(1.51)
" .. or~ " [d ( av~ ) av~ ]L-m,rj· -.- = L- - m,v~· -.-. -mjV,· -.- ,~ aqj ~ dt aq] aq)
and the second term on the left-hand side ofEq. (1.45) can be expanded into
Identifying L, !m~v~ with the system kinetic energy T, O'Alembert's principle
(cf. Eq. (1.45» becomes
(1.52)
Note that in a system of Cartesian coordinates the partial derivative of T \\ith
respect to q) vanishes. Thus, speaking in the language of differential geometry,
this term arises from the curvature of the coordinates qJ' In polar coordinates,
e.g., it is in the partial derivative of T with respect to an angle coordinate that the
centripetal acceleration term appears.
Thus far, no restriction has been made on the nature of the constraints other
than that they be workless in a virtual displacement. The variables qJ can be any
set of coordinates used to describe the motion of the system. If, however, the con-
straints are holonomic, then it is possible to find sets of independent coordinates
qJ that contain the constraint conditions implicitly in the transformation equations
(1.38). Any virtual displacement 8q] is then independent of oqk, and therefore the
only way for (1.52) to hold is for the individual coefficients to vanish:
d(aT) aT
dt aq) - aq) - Q}.
There are n such equations in all.
When the forces are derivable from a scalar potential function V,
(1.53)
1.4 D'Alembert's Prmciple and Lagrange's Equations
Then the generalized forces can be written as
11
'" ari '" arrQ) = L..,Fr • -.- = - L- \11 V· -.-,
r aqJ r aq]
which is exactly the same expression for the partial derivative of a function
- V (rt, r2, ... , rN, t) with respect to qr
av
Q} = - aq,'
Equations (1.53) can then be rewritten as
(1.54)
(1.55)~ (~:) _ a(~ - V) =0.
dt aqj aqr
The equations of motion in the fonn (1.55) are not necessanly restricted to conser-
vative systems, only if V is not an explicit function of time is the system conserva-
tive (cf. p. 4). As here defined. the potential V does not depend on the generalized
velocities. Hence, we can include a tenn m V in the partial derivative with respect
to q):
:!- (a(T ~ V») _ aCT - V) = o.
dt 8q] 8q)
Or. defining a new function, the Lagrangian L, as
L = T - V,
the Eqs. (1.53) become
d(aL) aL
dt aq} - oqJ = 0,
(1.56)
(1.57)
(1.57')
expressions referred to as "Lagrange's equations."
Note that for a particular set of equations of motion there is no unique choice
of Lagrangian .;;uch that F.q" (1 57) lead to the equations of motion in the given
generalized coordinates. Thus, in Derivations 8 and 10 itis shown that if L(q, q, t)
is an approximate Lagrangian and F(q, t) is any differentiable function of the
generalized coordinates and time, then
,. . dF
L (q, q, t) = L(q. q, t) + dt
is a Lagrangian also resulting in the same equations of motion. It is also often
pOSSIble to find alternative Lagrangians beside those constructed by this prescrip-
tion (see Exercise 20). While Eg. (1.56) is alwaysa suitable way to construct a
Lagrangian for a conservative system, it does not provide the only Lagrangian
suitable for the given system.
(1.58)
22 Chapter 1 Survey of the Elementary Principles
1.5 • VELOCITY-DEPENDENT POTENTIALS AND
THE DISSIPATION FUNCTION
Lagrange's equations can be put in the fonn (1.57) even if there is no potential
function, V, in the usual sense, providing the generalized forces are obtained from
a function U(qj' ti j ) by the prescription
au d (au)
Q) = - aq} + dE aqj .
In such case, Eqs. (1.57) still follow from Eqs. (1.53) with the Lagrangian given
by
L = T - U. (1.59)
Here U may be called a "generalized potential," or "velocity-dependent poten-
tial." The possibility of using such a "potential" is not academic; it applies to one
very important type of force field, namely. the electromagnetic forces on movmg
charges. Considering its importance, a digression on this subject is well worth-
while.
Consider an electric charge, q, of mass m moving at a velocity. v, in an other-
wise charge-free region containing both an electric field. E. and a magnetic field.
B, whIch may depend upon time and position. The charge experiences a force,
called the Lorentz force, given by
F = q[E+ (V x B)]. (1.60)
Both E(t, x, y, z) and B(t, x, y, z) are continuous functions of time and posinon
derivable from a scalar potential 4>(t, x, y, z) and a vector potentia] A(t, x, y. z)
by
and
aA
E=-V'q>--at
B=V' xA.
(1.6] a)
(1.61b)
The force on the charge can be derived from the following velocity-dependent
potential energy
U=q¢-qA.v,
so the Lagrangian, L = T - U, is
L = !mv2 - qtjJ +qA • v.
(1.62)
(1.63)
1.5 Velocity-Dependent Potentials and the Dissipation Function
Consldering just the x-component of Lagrange's equations gives
.. (aA~ aAy aAz) (a1> dAx )
mx = q Vx ax + Vy ax + Vz ax - q ax +dt .
23
(1.64)
The total time derivative of Ax is related to the particle time derivative through
dAx aAx
--=--+v.VAxdt at
aAx aAx aAx aAx
= -a- + vx -a- + vY-a- +vz-,:\-,t x Y oZ
Equation (1.61 b) gives
B) ( aAy aAx) (aA z aAx )(v x x = vy -a- - -a- + Vz - - -a- .x y ax z
(1.65)
Combining these expressions gives the equation of motion in the x-direction
mx = q [Ex + (v x B)x]. (1.66)
On a component-by·component comparison, Eqs. (1.66) and (1.60) are identical,
showing that the Lorentz force equation is derivable from Eqs. (1.61) and (1.62).
Note that if not all the forces acting on the system are derivable from a poten-
tial, then Lagrange's equations can alwa)'s be written in the fonn
where L contains the potential of the conservative forces as before, and QJ rep-
resents the forces not arising from a potential. Such a situation often occurs when
frictional forces are present. It frequently happens that the frictional force is pro-
portional to the velocity of the particle, so that its x-component has the fonn
Fj.\ = -kxvl"'
Frictional forces of this type may be denved in terms of a function F, known as
Rayleigh's dissipation function, and defined as
:F = ~L (kxv:x + kyv;y + kzv~),
I
(1.67)
where the summation is over the particles of the system. From this definition it is
clear that
aF
Ff =--
x a'Vx
24 Chapter 1 Survey of the Elementary Principles
or, symbolically,
(1.68)
(1.69)
by (1.51).
We can also give a physical interpretation to the dissipation function. The work
done lJy the system against friction is
dWf = -Ff •dr = -Ff • vdt = (kxv; + kyv~ - kzv;) dt.
Hence, 2F is the rate of energy dissipation due to friction. The component of (he
generalized force resulting from the force of friction is then given by
'"'" ar; '"'" ar;Q] = L-Fj,. 8q = - L- '\IvF· aq
I) I
L or,=- '\IF--vail; ,
aF
- - fJq)·
An example is Stokes' law, whereby a sphere of radius a moving at a speed
v, in a medium of viscOMty TJ experiences the frictional drag force F f = 61l'TJav.
The Lagrange equatIons with dissipation become
(1.70)!!.- (aL ) _ aL + aF =0
dt ail] aqJ ail] ,
so that two scalar functions, Land F, must be specified to obtain the equations
of motion.
1.6 • SIMPLE APPLICATIONS OF THE LAGRANGIAN FORMULATION
The previous sections show that for systems where we can define a Lagrangian,
i.e., holonomic &ystems with applied forces derivable from an ordinary or gen-
eralized potential and workless constraints, we have a very convenient way of
setting up the equations of motion. We were led to the Lagrangian formulation
by the desire to eliminate the forces of constraint from the equations of motion,
and in achieving this goal we have obtained many other benefits. In setting up the
original fonn of the equations of motion, Eqs. (1.19), it is necessary to work with
many vector forces and accelerations. Wih the Lagrangian method we only deal
with two scalar functions, T and V, which greatly simplifies the problem.
A straightforward routine procedure can now be established for all problems
of mechanIcs to which the Lagrangian formulation is applicable. We have only to
write T and V in generalized coordinates, form L from them, and substitute in
(1.57) to obtain the equations of motion. The needed transformation of T and V
from Cartesian coordinates to generalized coordinates is obtained by applying the
1.6 Simple Applications of the Lagrangian Formulation
transformation equations (1.38) and (1.45'). Thus, T is given in general by
25
It is clear that on carrying out the expansion, the expression for T in generalized
coordinates will have the fonn
(1.71)
where Mo. MJ , M jk are definite functions of the r's and t and hence of the q's
and t. In fact. a compari~on shows that
and
~ ar, 8ri
Mj = ~m,"7it. 8qj' (1.72)
Thus. the kinetic energy of a system can always be written as the sum of three
homogeneous functions of the generalized velocities,
T = 11) + T! + Tz, (1.73)
where To is independent of the generalized velocities. Tl is linear in the velocit!es.
and T2 is quadratic in the veJocities. If the transformation equations do not contain
the time explicitly, as may occur when the constraints are independent of time
(scleronomous), then only the last term in Eq. (1.71) is nonvanishing, and T is
always a homogeneous quadratic fonn in the generalized velocities.
Let us now consider simple examples of this procedure:
1. Single particle in space
(a) Cartesian coordinates
(b) Plane polar coordinates
2. Atwood's machine
3. Time-dependent constraint-bead sliding on rotating wire
1. (a) Motion of one particle: using Cartesian coordinates. The generalized
forces needed in Eq. (1.53) are obviously Fx , Fy , and Fz• Then
26 Chapter 1 Survey of the Elementary Prine pies
T= !m (X 2 + j2 + Z2),
8T = 8T = aT = 0,
ax 8)1 8z
8T .
ax = mx,
and the equations of motion are
d ( .
- mx) = Fx ,dt
aT .
aj = my,
rl .
-(my) = Fy ,dt
8T .
at = mz,
:t (mz) = Fz· (1.74)
We are thus led back to the original Newton's equations of motion.
(b) Motion ofone particle: using plane polar coordinates. Here we must ex-
press T in tenns of rand 8. The equations of transfonnation, i.e., Eqs. (1.38), in
this case are simply
x = rcos/i
y = rsin8.
By analogy to (1.46), the velocities are given by
i = r cos/i - rB sine,
y =r sine +re cose.
The kmetic energy T = !m(i2 + )12) then reduces formally to
(I.75)
An alternative derivation of Eq. (1.75) is obtained by recognizing that the plane
polar components of the velocity are r along r, and r8 along the direction per-
pendicular to ." denoted by the unit vector n. Hence, the square of the velocity
expressed in polar coordinates is simply ;'2. + (1'8)2. With the aid of the expression
A ~ ~
dr = rdr + r9de + kdz
for the differential position vector, dr, in cylindrical coordinates, where r and
8 are unit vectors in the r and 8-directions, respectively, the componentsof the
g~lI~lalil.eU furc~ can be ublained from the definition, Eq. (1.49),
or AQr =F· - =F·r= Fr ,ar
ar A
Qe = F· - = F· r8 = r F(J,ae
1.6 Simple Applications of the Lagrangian Formulation
FIGURE 1.6 Derivative of r with respect to ().
27
since the derivative of r with respect to 6 is, by the definition of a derivative, a
vector in the direction of e(cf. Fig. 1.6). There are two generalized coordinates,
cUll! Ll~ldult: twu Laglange equations. The derivative:> occuning in the r equation
are
aT '2
ar = mre ,
and the equation itself i ...
aT .
-=mrar . !!:... (aT) = mrdt af '
.. 0"2 r:'mr - mr = Cr,
the second tenn being the centripetal acceleration tenn. For the (J equation, we
have the derivatives
aT8ii = 0, d ( 2 .) 2" •dt mr e = mr (J + 2mrrfJ,
~o that the equation becomes
d ( 2') 2" .dr mr e = mr (J + 2mrNJ =rFe.
Note that the left side of the equation is just the time derivative of the angular
momentum, and the right side is exactly the applied torque, so that we have simply
rederived the torque equation (1.26), where L = mr2(j and N(f.) = rFe.
2. Atwood's machine-(See Fig. 1.7) an example of a conservative system
with holonomic. scleronomous constraint (thc pulley is assumed frictionless and
massless). Clearly there is only one independent coordinate x, the position of
the other weight being determined by the constraint that the length of the rope
between them is l. The potential energy is
28 Chapter 1 Survey of the Elementary Principles
---..,.-~
I-x
FIGURE 1.7 Atwood's machine.
while the kinetic energy is
T =! (MI + M2)X 2.
Combining the two, the Lagrangian has the form
L = T - V = ! (M) + M2) x2 + Mlgx + M2g(l - x).
There is only one equatIon of motion, involving the derivatives
8L
ax = (Ml - M2)g,
oL
ax = (Ml +M2)X,
so that we have
~MI + M2)X = (MI - M2) g,
or
.. Ml -M2
X = g,
Ml+M2
which is the familiar result obtained by more elementary means. This trivial prob-
lem emphasizes that the forces of constraint-here the tension in the rope-
appear nowhere in the Lagrangian fonnulation. By the same token, neither can
the tension in the rope be found directly by the Lagrangian method.
3. A bead (or ring) slzding on a uniformly rotating wire in a force-free space.
The wire is straight, and is rotated uniformly about some fixed axis perpendicular
to the wire. TIris example has been chosen as a simple illustration of a constraint
Derivations 29
being time dependent, with the rotation axis along z and the wire in the xy plane.
The transfonnation equations explicitly contain the time.
x = rcoswt.
y = rsinwt.
(w =angular velocity of rotation)
(r = distance along wire from rotation axis)
While we could then find T (here the same as L) by the same procedure used to
obtain (1.71), it is simpler to take over (1.75) directly, expre~sing the constraint
by the relation iJ = w:
T =!m (r 2 +r2( 2).
Note that T is not a homogeneous quadratic function of the generalized velocities,
since there is now an additional tenn not involving r. The equation of motion is
then
mr = mrw
2
= 0
or
., ?
r = rw-,
which IS the familiar simple harmonic oscillator equation with a change of sign.
The solution r = eCIJ / shows that the bead moves exponentially outward because
of the centripetal acceleration. Again, the method cannot furnIsh the force of con-
straint that keeps the bead on the wire. Equation (1.26) with the angular momen-
tum, L = m,.2w2eu)t. provides the force F = N/r, which produces the constraint
force, F = mrw2eClJt , acting perpendicular to the wire and the axis of rotation.
DERIVATIONS
1. Shliw that for a single parucle with constant mass the equation of motion imphec; the
following differential equation for the kinetic energy:
dT
- =F·v,
dl
while if the mass variec; with time the corresponding equation h
d(mT) = F. P
dt .
2. Prove that the magnitude R of the position vector for the center of mass from an
arbitrary origin is given by the equation
M 2R2 = MLm,r; - ~ Lm,m)r~.
I I I
30 Chapter 1 Survey of the Elementary Principles
3. Suppose a system of two particles is known to obey the equauons of motion, Eqs.
(1.22) and (1.26). From the equations of the motion of the individual particles show
that the internal forces between particles satisfy both the weak and the strong laws
of action and reaction The argument rna) be generalized. to a !:lystem with arbitrary
number of particles, thus proving the converse of the argumenL\; leading to Eqs. (1.22)
and (1.26).
4. The equations of constramt for the rolling dIsk, Eqs. (1.39), are special cases of gen-
eral lInear differential equations of constraint of the form
nL8, (Xl • ... , x ll )d:1C, = O.
1=1
A constramt condition of thi~ type is holonomIc only if an integrating function
I (x I ' ... , xn) can be found that turn~ it bto an exact differential. Clearly the func-
tion must be such that
for all i :f= j. Show that no such mtegrattng factor can be found for either of Eqs.
(1.39).
5. Two wheels of radius a are mounted on the ends of a common axle of length b such
thaL the wheels rotate independently. The whole combmatIon rolls without slipping on
a plane. Show that there are two nonholonomic equations of constraint,
cosOdx + sinOdy = 0
sinOdx - cosOd" = !a (d~ + d~') ,
(where e,~, and ~' ha\'e meanings similar to those in the problem of a single vertical
l1l:>k., and (x. y) are the coordinates of a point on the axle mIdway between the two
wheels) and one holonomic equation of constraint,
a ,
o= c - - (~ - ~ ).
b
where C is a constant.
6. A partIcle moves m the xy plane under the constramt that i~ velocity vector is al-
ways directed towards a point on the x axis whose absclllsa is some given function of
time I(t). Show that for I(t) differentiable, but otherwise arb.tracy, the constraint is
nor.holonomic.
7. Show that Lagrange's equations in the fonn of Eq~. (1.53) can also be written as
at aT
-a' -2-a = Ql'qj q)
These are somellmes known as the Nielsen form of the Lagrange equations.
8. If L is a Lagrangian for a system of n degrees of freedom satisfying Lagrange's equa-
tIons, show by direct substitution that
Exercises 31
aho ~atisfies Lagrange's equations where F is any arbitrary, but differentiable, func-
tion of itl> arguments.
9. The electromagnetic field is invanant under a gauge transfonnation of the scalar and
vector potential gIven by
A -+ A + V""(r, t),
"a""<P -+ <P - --,
c at
where"" IS arbItrary (but differentiable). \\that effect does this gauge transformation
have on the Lagrangian of a particle moving in the electromagnetic field? Is the motion
affected?
10. Let qJ' •••• qn be a set of independent generalized coordinates for a system of n
degrees of freedom, wi:h a Lagrangian L(q, q, t). Suppose we transfonn to another
set of mdependent coordmates St, ... , Sn by means of transformation equatIOns
l = 1, ... ,n.
(Such a transfonnatton i~ called a pomt transformation.) Show that If the Lagrangian
function is expressed as a functIon of S)' SJ' and t through the equations of transfOl-
manon, then L satisfies Lagrange's equatio'ls with respect to thes coordinates:
d(OL) aL
dt as) - as) = o.
In other words, the form of Lagrange's equations is invariant under a pomt transfor-
mation.
EXERCISES
11. ConMder a unifonn thin disk that rolls without slippmg on a horizontal plane. A hori-
zontal force is applied to the center of the dIsk and in a direction parallel to the plane
of tile disk.
(a) Denve Lagrange's equations and find the genenilized force.
(b) Discuss the motion If the force is not applied pandlel to the plane of the disk.
12. The escape velocity of a partlcle on Earth is the minimum velocity required at Earth's
sur-ace in order that the particle can escape from Earth's gravitational field. Neglecting
the resistanceof the atmosphere, the system i~ conservative. From the conservation
theurem fUl pulenl1<ll plu& kinetic energy l>how that the escape velocity for Earth,
Ignoring the presence of the Moon, is 11.2 kmls.
13. Rockets are propel1ed by the momentum reaction of the exhaust gases expelled from
the tail. Since these gases anse from the reaction of the fuels carried in the rocket, the
ma..,s of lhe rocket is not conMant, but decreases a~ the fuel is expended. Show that the
eql.:ation of motion for a rocket projected vertically upward in a uniform gravitational
32 Chapter 1 Survey of the Elementary Principles
field, neglecting atmospheric friCtlon, IS
dv ,dm
m-=-v--mgdt dt '
where m I~ the mass of the rocket and v' is the velocity of the escaping gases relative to
the rocket. Integrate this equation to obtain v as a function ofm, assuming a cons'ant
time rate of loss of mass. Show, for a roclet starting imtially from re~t, with v' equal
to 2.1 rn/s and a mass lo~s pel second equal to IJ60th of the initial mass, that in order
to reach the escape velocity the ratio of the weight of the fuel to the weight of the
empty rocket must be almost 300!
14. Two points of m&~m are Jomed by a ngld weightle"ls rod of length l, the center of
\\Ihich is constrained to move on a clTcle of radius a. Express the kInetic energy m
generahzed coordmates.
15. A point particle moves in space under the influence of a force derivable from a gener-
alized potential of the fonn
U(I,V)= V(r)+u·L.
where r is the radius vector from a fixed point, L IS the angular momentum about that
point, and u is a fixed vector in space.
(al Fmd the components of the force on the particle m both CartesIan and "lphencal
polar coordmates, on the baSIS of Eq. 11.58).
(b, Show that the component,> m the two coordinate systemf) ue related to each other
as m Eq. (1.49).
(c) Obtain the equatIOns of motIon in c;phencal polar coordinate,>.
16. A particle moves in a plane under the influence of a force, acting toward a center of
force, whose magniluce IS
_ 1 ( ,.2 - 2,,)F--2 1- ? 'r c-
where r IS the distance of the particle to the center of force. Find the generaliled
potential that will result in such a force, and from that the Lagrangtan for the motion
10 a plane. (The expression for F represents the force between lwo chargef) in Weber's
electrodynamics.)
17. A lluc1eus. originally at rest, decays radio~ct1Vely by eJllitting an electron of momen-
tum 1.73 MeVIc, and at right angle'> to the direction of the electron a neutrmo \\11th
momentum 1.00 MeV/c. (The MeV, milhon electron volt, is a unit of energy used
m modern phySICS, equal to 1.60 x 10-13 J. Correspondingly, MeVk IS a uwt of
lInear momentum equal to 5.34 x 10-22 kg.rn/s.) In what dlrection does the nu-
clem; recoil? What is -ts momentum In MeV/c? If the ma,>~ of the residual nucleus
is 3.90 x JO-25 kg what is its kinetic energy. in electron volts?
18. A Lagrangian for a particular physical system can be wntten as
L' = '; (ai 2 + 2biy +ci) - ~ (ax2 +2bxy + cy2) ,
where a, b. and c are arbitrary con~tants but ~ubJect to the condillon that b2 - ac :f: O.
Exercises 33
What are the equations of motion? Examme particularly the two cases a = 0 = c
and b = 0, c = -a. What is the physical system described by the above Lagrangian?
Show that the usual Lagrangian for this system as defined by Eq. (1.57') is related
to L
'
by a point transformation (cf. Derivation 10). What is the significance of the
condlbon on the value of b2 - ac?
19. Obtain the Lagrange equations of motion for a ~pherical pendulum, Le., a ma~s point
sUIlpended by a rigid weightless rod.
20. A particle of mass m moves in one dimension such that It has the Lagrangian
m2i 4
L = - + m.r2V(x) - V2(X),
12
where V is some differentiable function ofx. Find the equatIOn of motion for x(t) and
describe the phyMcal nature of the system on the basis of this equation
21. Th.o mass points of mass ml and m2 are connected by a stnng passing through a
hole in a smooth table so thal m 1 rests on the table surface and m2 hangs ~u~pended.
A~l1mine m2 move" only in a vertical line. what are the ieneralized coordinates for
the system? Write the Lagrange equations for the system and, if possible, mscusf)
the physical sIgnificance any of them might have. Reduce the' problem to a single
second-order dIfferentIal equation and obtain a first lDtegral of the equation. Wha: is
its physical SIgnificance? (Consider the motion only until m1 reaches the hole.)
22. Obtam the LagrangIan and equations of motion for the double ;>endulum Illustratec In
Fig 1.4, where the lengths of the pendula are II and 12 WIth corresponding masses m I
and m2.
23. Obtain the equation of motion for a particle falling vertically under the influence of
gravity when frictionalforce~ obtainable from a dissipation function !kv2 are present.
Integrate the equation to obtain the velocity as a function of arne and show that the
maxmmm possible velocity for a fall from rest is v = mgjk.
24. A spring of rest lengtt> La (no tension) is connected to a support at one end and has
a mass M attached at (he other. Neglect the mass of the spring, the dimension of the
mass M, and aSSUme that the motion IS confined to a vertical plane. Also, assume that
the spring only stretches without bendmg but it can swing In the plane.
(a) Usmg the angular displacement of the mass from the vertical and the length that
the string has stretched from its rest length (hanging with the mass m), find La-
granee',\ eqllation~.
(b) Solve these equatIons for small stretching and angular displacemenU>.
(cJ Solve the equatIons in part (a) to the next order in both stretching and angular
dillplacement. This part is amenable to hand calculations. Using some reasonable
assumptions about the spnng con~tanl, the mass, and the rest length. disc~s the
motion. If) a resonance likely under the assumptions statec. in the problem?
(d) (For analytic computer programs.) ConSIder the spring to have a total mass
In « M. Neglecting the bending of the spring. set up Lagrange's equations
correctly to first order in m and the angular and linear dIsplacements.
(e) (For numencal computer analysIs.) Make sets of rea.~onable assumptlons of the
constants In part (a) and make a single plot of the two coordinates as functions of
time.
CHAPTER
2 Variational Principles andLagrange's Equations
2.1 • HAMILTON'S PRINCIPLE
The derivation of Lagrange's equations presented in Chapter I started from a
consideration of the instantaneous state of the system and s~all virtual displace-
ment~ about the instantaneous state, Le., from a "differential principle" such as
D'Alembert's principle. It is also possible to ohtain T~grange's equations from a
principle that considers the entire motion of the system between times tl and 12,
and small virtual variations of this motion from the actual motion. A principle of
this nature is known as an "integral principle."
Before presenting the integral principle, the meaning attached to the phrase
"motion of the system between times tl and t2" must first be stated In more pre-
cise language The in<:t~ntaneol1s configuration of a system is described by the
values of the n generalized coordinates qI, ... , qn, and corresponds to a particu-
lar pomt in a Cartesian hyperspace where the q's form the n coordinate axes. This
n-dimensional space is t1erefore known as configuration space. As time goes on,
the state of the system changes and the system point moves in configuration space
tracing out a curve, desc:ibed as "the path of motion of the system." The "motlon
of the system," as used above, then refers to the motion of the system point along
this path in configuratio"l spac.e. Time can be considered formally as a parame-
ter of the curve; to each point on the path there is associated one or more values
of the time. Note that configuration space